What advice would you give to young people starting out in math (i.e., high school students and young researchers)?

I am really not in a position to give advice; I usually use the career advice on [Australian mathematician] Terry Tao's web page for myself! Also, everyone has a different style, and something that works for one person might not be so great for others.

What advice would you give lay persons who would like to know more about mathematics—what it is, what its role in our society has been and so on? What should they read? How should they proceed?

This is a difficult question. I don't think that everyone should become a mathematician, but I do believe that many students don't give mathematics a real chance. I did poorly in math for a couple of years in middle school; I was just not interested in thinking about it. I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers.

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The quote you may have seen already is Mirzakhani's response to the second half of that two-part question, i.e. the advice she would give to lay persons. It's great advice. The bit about the beauty of things revealing themselves only to their most patient followers is as true of mathematics as it is of many pursuits in life. But her words carry more weight, I think, when they're preceded by her response to the first question.

It's reassuring to learn that one of the world's greatest mathematicians reads another mathematician's blog for career advice. It's refreshing to hear her verify that yes, there are different learning styles, different passions, and different skill sets, and that "something that works for one person might not be so great for others." She is a teacher, remember. A Stanford professor who oversees not only an introductory course on ergodic theory (one of her research interests), but advanced courses as well. This also helps contextualize her belief that "that many students don't give mathematics a real chance" – an observation she makes not only as someone who once struggled with mathematics herself, but as a teacher of students. Students she recognizes may not become mathematicians (or chemists, or physicists, or biologists, or anthropologists, or poets, or literary theorists), but who might still invest the time necessary to access, appreciate, and celebrate the world's more subtle beauties.